Our primary application will be to the study of distributions in theories with \(\mathrm {SOP}_n\)  for \(n\ge 2\)  (defined in [1]}). In particular, the compatibility of conditions coming from an \(\mathrm {SOP}_2\) -type (as defined in [2]}) has the same structure as graphs arising from the compatibility relation on finite trees, and the compatibility of conditions that arise from types indicating a cut within an ultraproduct of infinite linear orders has the structure of graphs where the vertices are intervals in the order (or any infinite linear order) and the edges indicate non-empty intersections, allowing a characterization of the distributions for these sorts of types in terms of the corresponding classes of graphs. Moreover, since we know from [1]} and [2]} that an ultrafilter \( is good if and only if \)  \(\lambda ^+\) -saturates structures having \(\mathrm {SOP}_2\) , we find a characterization in Theorem REF  of good ultrafilters both in terms of whether a relatively small class of distributions has multiplicative refinements and in terms of whether ultraproducts of comparability graphs of finite trees allow small complete subgraphs to have internal complete extensions.
